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  "body": "\n## Statement\n\nLet $p$ be a prime, let $\\Z_p$ be the ring of [[$p$-adic integers|GBN9F8FAQ3]], and let $\\Q_p$ be the field of [[$p$-adic numbers|CHT5JYM6QW]].\nFor $k \\in \\N$ and $x \\in \\Z_p$ let $\\binom{x}{k} = \\frac{x(x-1)\\cdots(x-k+1)}{k!}$.\nLet $f \\colon \\Z_p \\to \\Q_p$ be a [[continuous function|0P27W9RW0G]], and let $a_k = \\Delta^k f(0)$ be its [[$k$-th forward difference|E38801KVJ1]] at $0$ with step $1$.\nThen $a_k \\to 0$ in $\\Q_p$, and\n$$f(x) = \\sum_{k=0}^{\\infty} a_k \\binom{x}{k} \\quad \\text{for all } x \\in \\Z_p,$$\nwhere the series [[converges uniformly|08FPABDQ4C]] on $\\Z_p$.\nConversely, for every sequence $(a_k)$ in $\\Q_p$ with $a_k \\to 0$, this series defines a continuous function $f \\colon \\Z_p \\to \\Q_p$ with $\\Delta^k f(0) = a_k$ for all $k$.\n\n## Remarks\n\nThe binomial polynomials $\\binom{x}{k}$ take integer values on $\\N$, which is dense in $\\Z_p$, so they map $\\Z_p$ into $\\Z_p$; hence $\\abs{\\binom{x}{k}}_p \\le 1$ and the series converges uniformly when $a_k \\to 0$.\nOn $\\N$ the expansion is [[Newton's forward difference formula|DDHCMKHD2T]], and both sides are continuous, so the main point is that $a_k \\to 0$.\nThis is proved by showing that $\\Delta^{p^m}$ is small on continuous functions for large $m$, using the expansion of $\\Delta^{p^m}$ in shifts and the divisibility of binomial coefficients by $p$.\nMoreover, $\\sup_{x \\in \\Z_p} \\abs{f(x)}_p = \\max_k \\abs{a_k}_p$, so the functions $\\binom{x}{k}$ form an orthonormal basis of the Banach space of continuous functions $\\Z_p \\to \\Q_p$.\nExample: for $u \\in \\Z_p$ with $\\abs{u - 1}_p \u003c 1$, the function $x \\mapsto u^x$ is continuous, and its expansion is $\\sum_k (u - 1)^k \\binom{x}{k}$, with $a_k = (u-1)^k \\to 0$.\n\n## Explanations\n\n### L1 human\n\nEvery continuous function on the $p$-adic integers is given by a Newton series that converges everywhere.\nThe coefficients are the forward differences at $0$, which can be read off from the values $f(0), f(1), f(2), \\dots$.\nOver the real numbers nothing like this holds, since a continuous function on $\\R$ is far from being fixed by its values at the natural numbers.\n\n### L2 human\n\nThe theorem identifies the space of continuous functions $\\Z_p \\to \\Q_p$ with the space of null sequences, isometrically for the sup norm.\nIt is a basic tool in $p$-adic analysis, for example to construct $p$-adic interpolations of functions defined on the natural numbers, and to describe $p$-adic measures by their moments.\nIt is the $p$-adic counterpart of the [[Newton series|Y2HXSEA2EE]] expansion, where in the real case convergence is a delicate question.\n",
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